Express the following declarative sentences in propositional logic; in each case state what your respective propositional symbols p$,$ q$,$ etc. mean:

a$)[2\%]$ The contract is not valid unless it has been signed by all parties and validated by a public authority.

b$)[2\%]$ Tomorrow we will either visit the museum or walk down the pier, but not both. c$)[2\%]$ If the parliament approves the no$-$confidence motion, the government must resign or ask for a snap election.

For each of the following invalid inferences, give examples of declarative sentences in natural language for the atoms p and q$,$ such that the premises are true, but the conclusion false.

d$)[2\%]$ p $$ q $$ p $\backslash$Lambda q

e$)[2\%]$ p $->$ q $$ p $$ q